Global disorder transition in the community structure of large-q Potts systems

TL;DR

This paper investigates how increasing the number of communities in large networks induces a global disorder transition in community detection, linking it to a Potts model analogy and identifying a crossover temperature for solvability.

Contribution

It introduces a theoretical framework connecting community detection disorder transitions to Potts models, providing bounds and conditions for solvability in large systems.

Findings

Large-q systems tend toward global disorder above a certain temperature

A crossover temperature $T_\times$ predicts the transition from solvable to unsolvable phases

Local algorithms can still find solutions despite global disorder transitions

Abstract

We examine a global disorder transition when identifying community structure in an arbitrary complex network. Earlier, we illustrated [Phil. Mag. 92, 406 (2012)] that "community detection" (CD) generally exhibits disordered (or unsolvable) and ordered (solvable) phases of both high and low computational complexity along with corresponding transitions from regular to chaotic dynamics in derived systems. Using an exact generalized dimensional reduction inequality, multivariate Tutte polynomials, and other considerations, we illustrate how increasing the number of communities q emulates increasing the heat bath temperature T for a general weighted Potts model, leading to global disorder in the community structure of arbitrary large graphs. Dimensional reduction bounds lead to results similar to those suggested by mean-field type approaches. Large systems tend toward global insolvability in the limit of large q above a crossover temperature $T_\times\approx L|J_e|/[N\ln{} q]$ where |J_e| is a typical interaction strength, L is the number of edges, and N is the number of nodes. For practical system sizes, a solvable phase is generally accessible at low T. The global nature of the disorder transition does not preclude solutions by local CD algorithms (even those that employ global cost function parameters) as long as community evaluations are locally determined.